Why is the differential of a constant zero?

Any constant (say k) can be rewritten as kx⁰ since x⁰=1. When you differentiate this, the 0 which is the power of the x term gets dragged to the front and is multiplied with the rest of the expression (according to the short-hand differential method). So if f(x) = kx⁰ is differentiated, we get f '(x) = 0kx^-1 = 0.